SeedRandom[1]
region = ConvexHullMesh[RandomReal[{-1, 1}, {50, 2}]]

sol = Minimize[Indexed[x, 2], x ∈ RegionBoundary@region]
point = x /. Last@sol
(* {-0.976044, {x -> {0.579608, -0.976044}}} *)
(* {0.579608, -0.976044} *)

(* plot the solution: *)
Show[region, Epilog -> {Point@x /. Last@sol}]